For some reason it, I need say "I don't have any toothpaste", instead of simply saying "I don't have toothpaste"
Is zębów here a plural? And what grammatical case is it in?
I do not think the "pasty", in this case, is plural. I believe it is "pasty" because the negated accusative verb (mice) takes on genitive and genitive for "pasta" is "pasty". Is this correct?
Yes, you're right. But that comment was about "zębów" :)
I'm confused by 'mice', I guess it's a typo plus lack of Polish signs of "mieć"?
Pasta do zębów looks like "pasta de dente" in Portuguese
how's the declension for pasty do zębów like?
https://en.wiktionary.org/wiki/pasta#Noun_23 - that's the declension for "pasta", and then "do zębów" will always stay unchanged, as it's a prepositional phrase.
Eu não tenho pasta de dentes